The quasistatic approximation can be understood through the idea that the sources in the problem change sufficiently slowly that the system can be taken to be in equilibrium at all times.
This approximation can then be applied to areas such as classical electromagnetism, fluid mechanics, magnetohydrodynamics, thermodynamics, and more generally systems described by hyperbolic partial differential equations involving both spatial and time derivatives.
In the strict acceptance of the term the quasistatic case corresponds to a situation where all time derivatives can be neglected.
In classical electromagnetism, there are at least two consistent quasistatic approximations of Maxwell equations: quasi-electrostatics and quasi-magnetostatics depending on the relative importance of the two dynamic coupling terms.
To first order, the mistake of using only Biot–Savart's law rather than both terms of Jefimenko's magnetic field equation fortuitously cancel.